What is the OLD? I'd be surprised a Sturmey hub would be too wide for any modern dropouts even if you have the long axle already on there. Also keep in mind the cog on these hubs are dished which gives you 2mm or so of chainline adjustment.

The OLD was 134-5mm and the dropouts are 125mm.

So you're saying that I'll be able to adjust the rim from left to right easily by changing how much I screw the spokes in on each side?

Yes that is how it is done. When you tighten the spokes that go to the left flange the rim moves left, tighten the right side and it moves right. A full turn on a nipple can effect a surprisingly drastic change in the rim location.

This is my favorite wheelbuilding reference article. I keep it on the screen everytime I lace a wheel just in case I have a brain fart. https://www.sheldonbrown.com/wheelbuild.html

read it all the way through if you haven't before, there's a lot of good tips and insight in there.

Just keep in mind. This is not difficult, you'll be fine.

Are you sure? That seems way off to me. I'd be surprised if it was 120. Most Sturmey hubs were spaced at 110 with the short axle & you could get it up to 120 with the long axle.

+1. Building wheels is one of those things where doing it is easier than thinking about doing it. So just do i! Since your rim has offset spoke holes, make sure they're on the left side of the wheel, and go for it. Lacing it up is one process, truing it is another one. Take it one step at a time; once you have it all laced up, check all the spokes for accuracy before you start tightening them up. Don't worry, have fun... take your time and get it right, and you will be so proud of your creation you'll get addicted to wheelbuilding!

Just FYI, if you ended up centering both flanges between drops, you'd do better not to use an offset rim. The reason for an offset rim is to make spoke length on the left and right side more equal by compensating for the unevenness of the flange placement on the rear hub. The rim is built with the spoke bed off center, so the drive side flange is a further reach, and the non drive flange is a shorter reach for the spokes.

O/C rims are the great equalizer for the unevenness that goes on with rear wheels. Spoke tension on the non drive side is much closer to drive side tension on an o/c wheel than on a wheel with a centered rim. All new multi-speed rear wheels should use them.

But if you centered the hub flanges between the drops (not that you should), using an o/c rim would require you to build a wheel with some dish even though you could and should build it dishless.

As for spoke length, rear wheels typically have a 1-2mm difference in drive to non-drive side. With centered rims, left and right are most commonly 2mm apart in spoke length. With o/c rims, a 1mm difference between ds and nds lengths is most common, with the left still slightly longer. Calculations vary among hubs and rims. A 4mm offset is pretty typical with o/c rims. If I had to guess, I'd probably enter a three and call it a day. Other folks are right that certain dimensions don't change your spoke length all that much. ERD and flange diameter are by far the biggest determinants.

Finally, with a 1mm difference in spoke calc results you can use one length no problem. Some spoke manufacturers sell only in 2mm increments. In your case, I'd probably use 292mm on both sides.